AbstractThe global conformal gauge is playing the crucial role in string theory providing the basis for quantization. Its existence for two-dimensional Lorentzian metric is known locally for a long time. We prove that if a Lorentzian metric is given on a plain then the conformal gauge exists globally on the whole $${{\mathbb {R}}}^2$$ R 2 . Moreover, we prove the existence of the conformal gauge globally on the whole worldsheets represented by infinite strips with straight boundaries for open

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... d closed bosonic strings. The global existence of the conformal gauge on the whole plane is also proved for the positive definite Riemannian metric.